About our research

Analysis and numerics of multiphysics systems in acoustics

Coupled systems in bio-acoustics and elasto-acoustics arise as models in important medical applications, ranging from contrast imaging and tumor therapy to targeted drug delivery. The goal of the project is to perform analytical investigations of the underlying multiphysics models involving evolution PDEs and develop their efficient numerical approximation.

Stochastic optimization methods in machine learning

Stochastic gradient descent is the key tool for machine learning and large scale optimisation.
An interesting problem is how to efficiently adapt the time step in the algorithm to ensure convergence and to explore potential minima. This project would examine and compare taming, adaptivity and other ideas to control stability of the simulations.

Mathematical biology

We aim to model and analyse phenomena in the biosciences: for instance,  biofilm growth, cancer cell invasion or molecular cell signalling mechanisms. Depending on the concrete application the models are formulated as systems of ODEs, PDEs and/or stochastic ODEs or PDEs. Projects can have a theoretical focus or be interdisciplinary in collaboration with research groups at the Faculty of Science or the Radboud UMC.

Shape uncertainty quantification for neuroscience

The geometry of the brain has important consequences on its activity, but it is subject to uncertainty, due for example to variability among individuals and approximations inherent to imaging techniques. In this project in collaboration with the Vrije Universiteit Amsterdam, we aim at establishing the mathematical and numerical framework to quantify the effect of geometric uncertainty of the brain on neuronal activity.

Designing microstructure for functional materials

Materials have microstructures that determines their macroscopic behaviour. Despite the information is at the microscopic level, it is usually more convenient to have a macroscopic description of the the situation. The goal of the project is to rigorously get such a macroscopic description, and to use this knowledge to design composite materials with specific properties

Singularity theorems via Optimal Transport theory

The singularity theorems in General Relativity (going back to Roger Penrose and Stephen Hawking) show that singularities are inevitable at the Big Bang and inside Black Holes. Energy conditions that are formulated geometrically as lower Ricci curvature bounds are a major ingredient in these results. In recent years a synthetic theory for timelike and null Lorentzian Ricci curvature lower bounds has been developed with the help of Optimal Transport theory. In this project you learn and compare definitions for Lorentzian synthetic spaces and recast/reproof the singularity theorems in this new framework.

Quantum symmetric pairs in quantum group

The goal of the project is to investigate quantum symmetric pairs in quantum groups, which can be thought of as the appropriate analogue of symmetric spaces, such as spheres, hyperboloids, etc. An important application of this theory is the ability to perform noncommutative harmonic analysis on quantum symmetric spaces, by using both algebraic techniques as well as operator algebras. By further using tools from orthogonal polynomials and special functions and their interplay with representation theory of  quantum groups we want to understand the quantum symmetric pairs.

Survival analysis, time varying covariable

We aim to model the effect of a time varying covariate on a survival outcome. For example, a marker which can be measured in blood and the recurrence of cancer. Methods have been developed for single persons and one event, but not for twin data, recurrent events, competing risks, delayed entry.

Fractal percolation

Fractal percolation is a random fractal subset of the unit square. It is generated by repeatedly subdividing and randomly selecting subsquares on smaller and smaller scales. The goal of the project is to understand the limiting fractal set. In particular, the dependence of its structure on the parameters of the problem.

Automorphisms of regular trees

An infinite tree in which all vertices has the same degree has a huge group of automorphisms. We will look at groups of automorphisms that act transitively on edges and on infinite lines in the trees. Such groups have a surprisingly rich structure, with many subgroups that together form a BN-pair. In this way one extracts from pure geometry something very similar to an algebraic group, namely to SL(2,Q). The goal of the project is to prove statements of this kind with elementary techniques. 

Higher-Degree Points on Algebraic Curves

Which systems of polynomial equations with integer coefficients (so-called Diophantine equations) admit only finitely many solutions? This classical question lies at the heart of arithmetic geometry. Faltings' celebrated theorem asserts that any algebraic curve of genus at least two has only finitely many rational points. But what if we broaden the class of solutions? For instance, when does a curve of genus at least two have infinitely many points defined over quadratic number fields (or cubic, or quartic,....)? In this project, we explore such higher-degree solutions, investigating the conditions under which Diophantine equations admit infinitely many points of bounded degree, and how these results connect to deep geometric and arithmetic properties of the underlying curves.

The research within this group is characterised by its interdisciplinary aspect. There are close links to the groups in Applied Stochastics and Mathematical Physics as well as to other research institutes at Radboud University. This group focusses both on the theoretical and numerical aspects of the problems, and currently specialises in the following themes: Applied Analysis, Modelling and Applications, and Numerics and Computation.

Applied Mathematics

This group focusses on modern mathematical physics, especially on of the two great fundamental theories of physics: namely general relativity and quantum (field) theory. Their interest lies in both the pure and applied mathematical topics needed here, but also in applications to the frontiers of fundamental physics. There is a strong collaboration with the neighboring departments of astrophysics and high-energy physics.

Mathematical Physics

For many complex systems in nature and society, randomness provides an efficient description with great explanatory and predictive power. Stochastics is the area of mathematics that deals with processes and objects where randomness plays a role.

Probability and Statistics

Pure Mathematics is a very broad area that includes many subfields. Among the main directions that are represented in this research group are the following: Algebraic and Arithmetic Geometry, Algebraic Topology, Differential Geometry, Logic and Computer Algebra, Number Theory, and Representation Theory and Lie Theory.

Pure Mathematics